企业网站引导页模板,做seo比较好的网站,网站建设维护是啥意思,wordpress小工具友情链接本文仅供学习使用#xff0c;总结很多本现有讲述运动学或动力学书籍后的总结#xff0c;从矢量的角度进行分析#xff0c;方法比较传统#xff0c;但更易理解#xff0c;并且现有的看似抽象方法#xff0c;两者本质上并无不同。 2024年底本人学位论文发表后方可摘抄 若有… 本文仅供学习使用总结很多本现有讲述运动学或动力学书籍后的总结从矢量的角度进行分析方法比较传统但更易理解并且现有的看似抽象方法两者本质上并无不同。 2024年底本人学位论文发表后方可摘抄 若有帮助请引用 本文参考 黎 旭,陈 强 洪,甄 文 强 等.惯 性 张 量 平 移 和 旋 转 复 合 变 换 的 一 般 形 式 及 其 应 用[J].工 程 数 学 学 报,2022,39(06):1005-1011. 食用方法 质量点的动量与角动量 刚体的动量与角动量——力与力矩的关系 惯性矩阵的表达与推导——在刚体运动过程中的作用 惯性矩阵在不同坐标系下的表达 机构运动学与动力学分析与建模 Ch00-2质量刚体的在坐标系下运动Part4 2.2.4 牛顿-欧拉方程 Netwon-Euler equation 2.3 惯性矩阵的转换 Inertia-Matrix Transformation2.4 惯性矩阵的主轴定理} Principal Axis Theorem 对 H ⃗ Σ M / O F \vec{H}_{\Sigma _{\mathrm{M}}/\mathrm{O}}^{F} H ΣM/OF进一步处理可得 H ⃗ Σ M / O F ∑ i N m P i ⋅ R ⃗ O P i F × ( ω ⃗ F × R ⃗ O P i F ) ∑ i N m P i ⋅ R ⃗ O P i F × ( − R ⃗ O P i F × ω ⃗ F ) ∑ i N m P i ⋅ R ⃗ ~ O P i F ( − R ⃗ ~ O P i F ) ω ⃗ F \vec{H}_{\Sigma _{\mathrm{M}}/\mathrm{O}}^{F}\sum_i^N{m_{\mathrm{P}_{\mathrm{i}}}\cdot \vec{R}_{\mathrm{OP}_{\mathrm{i}}}^{F}\times \left( \vec{\omega}^F\times \vec{R}_{\mathrm{OP}_{\mathrm{i}}}^{F} \right)}\sum_i^N{m_{\mathrm{P}_{\mathrm{i}}}\cdot \vec{R}_{\mathrm{OP}_{\mathrm{i}}}^{F}\times \left( -\vec{R}_{\mathrm{OP}_{\mathrm{i}}}^{F}\times \vec{\omega}^F \right)}\sum_i^N{m_{\mathrm{P}_{\mathrm{i}}}\cdot \tilde{\vec{R}}_{\mathrm{OP}_{\mathrm{i}}}^{F}\left( -\tilde{\vec{R}}_{\mathrm{OP}_{\mathrm{i}}}^{F} \right)}\vec{\omega}^F H ΣM/OF∑iNmPi⋅R OPiF×(ω F×R OPiF)∑iNmPi⋅R OPiF×(−R OPiF×ω F)∑iNmPi⋅R ~OPiF(−R ~OPiF)ω F。进而得出 ⇒ [ I ] ∑ i N m P i ⋅ R ⃗ ~ O P i F ( − R ⃗ ~ O P i F ) \Rightarrow \left[ I \right] \sum_i^N{m_{\mathrm{P}_{\mathrm{i}}}\cdot \tilde{\vec{R}}_{\mathrm{OP}_{\mathrm{i}}}^{F}\left( -\tilde{\vec{R}}_{\mathrm{OP}_{\mathrm{i}}}^{F} \right)} ⇒[I]∑iNmPi⋅R ~OPiF(−R ~OPiF)
2.2.4 牛顿-欧拉方程 Netwon-Euler equation
刚体动力学中常用 { F ⃗ Σ M F m t o t a l ⋅ a ⃗ G F M ⃗ Σ M / G F [ I ] Σ M / G F α ⃗ M F ω ⃗ M F × ( [ I ] Σ M / G F ⋅ ω ⃗ M F ) \begin{cases} \vec{F}_{\Sigma _{\mathrm{M}}}^{F}m_{\mathrm{total}}\cdot \vec{a}_{\mathrm{G}}^{F}\\ \vec{M}_{\Sigma _{\mathrm{M}}/\mathrm{G}}^{F}\left[ I \right] _{\Sigma _{\mathrm{M}}/\mathrm{G}}^{F}\vec{\alpha}_{\mathrm{M}}^{F}\vec{\omega}_{\mathrm{M}}^{F}\times \left( \left[ I \right] _{\Sigma _{\mathrm{M}}/\mathrm{G}}^{F}\cdot \vec{\omega}_{\mathrm{M}}^{F} \right)\\ \end{cases} {F ΣMFmtotal⋅a GFM ΣM/GF[I]ΣM/GFα MFω MF×([I]ΣM/GF⋅ω MF)
2.3 惯性矩阵的转换 Inertia-Matrix Transformation
对于空间中的运动刚体而言刚体的惯性矩阵一般会根据运动坐标系 { M } \left\{ M \right\} \,\, {M}的基矢量为基底进行计算而不会直接考虑运动刚体在固定坐标系 { F } \left\{ F \right\} \,\, {F}下的惯性矩阵。此时运动坐标系 { M } \left\{ M \right\} \,\, {M}下计算得出的惯性矩阵记为 [ I ] M \left[ I \right] ^M [I]M。若运动坐标系 { M } \left\{ M \right\} \,\, {M}与固定坐标系 { F } \left\{ F \right\} \,\, {F}的基矢量满足 [ i ⃗ M j ⃗ M k ⃗ M ] [ Q M F ] T [ I ^ J ^ K ^ ] \left[ \begin{array}{c} \vec{i}^M\\ \vec{j}^M\\ \vec{k}^M\\ \end{array} \right] \left[ Q_{\mathrm{M}}^{F} \right] ^{\mathrm{T}}\left[ \begin{array}{c} \hat{I}\\ \hat{J}\\ \hat{K}\\ \end{array} \right] i Mj Mk M [QMF]T I^J^K^ 其中 [ Q M F ] T \left[ Q_{\mathrm{M}}^{F} \right] ^{\mathrm{T}} [QMF]T为转换矩阵Transition Matrix为正交矩阵Orthogonal Matrix满足 [ Q M F ] T [ Q M F ] − 1 [ Q F M ] \left[ Q_{\mathrm{M}}^{F} \right] ^T\left[ Q_{\mathrm{M}}^{F} \right] ^{-1}\left[ Q_{\mathrm{F}}^{M} \right] [QMF]T[QMF]−1[QFM] [ Q M F ] \left[ Q_{\mathrm{M}}^{F} \right] [QMF]又称旋转矩阵Rotation~Matrix 一个向量乘以一个正交阵相当于对这个向量进行旋转。也揭示了该矩阵的两个作用基底转换转换矩阵 [ Q M F ] T \left[ Q_{\mathrm{M}}^{F} \right] ^{\mathrm{T}} [QMF]T与向量旋转旋转矩阵 [ Q M F ] \left[ Q_{\mathrm{M}}^{F} \right] [QMF]则考虑最开始的图有 R ⃗ P i F R ⃗ M F [ Q M F ] R ⃗ P i M \vec{R}_{\mathrm{P}_{\mathrm{i}}}^{F}\vec{R}_{\mathrm{M}}^{F}\left[ Q_{\mathrm{M}}^{F} \right] \vec{R}_{\mathrm{P}_{\mathrm{i}}}^{M} R PiFR MF[QMF]R PiM
进而分析惯性矩阵若 O O O 点与固定坐标系原点 F F F 重合则有 [ I ] Σ M F ∑ i N m P i ⋅ [ ( R ⃗ P i F ) T R ⃗ P i F ⋅ E − R ⃗ P i F ( R ⃗ P i F ) T ] ∑ i N m P i ⋅ [ ( R ⃗ M F [ Q M F ] R ⃗ P i M ) T ( R ⃗ M F [ Q M F ] R ⃗ P i M ) ⋅ E − ( R ⃗ M F [ Q M F ] R ⃗ P i M ) ( R ⃗ M F [ Q M F ] R ⃗ P i M ) T ] { m t o t a l ⋅ [ ( R ⃗ M F ) T R ⃗ M F ⋅ E − R ⃗ M F ( R ⃗ M F ) T ] ⏟ [ I 1 ] Σ M F [ Q M F ] ( ∑ i N m P i ⋅ [ ( R ⃗ P i M ) T R ⃗ P i M ⋅ E − R ⃗ P i M ( R ⃗ P i M ) T ] ) [ Q M F ] T ⏟ [ I 2 ] Σ M F m t o t a l ⋅ [ ( R ⃗ M F ) T ( [ Q M F ] R ⃗ C o M M ) ⋅ E − R ⃗ M F ( [ Q M F ] R ⃗ C o M M ) T ] ⏟ [ I 3 ] Σ M F m t o t a l ⋅ [ ( [ Q M F ] R ⃗ C o M M ) T R ⃗ M F ⋅ E − ( [ Q M F ] R ⃗ C o M M ) ( R ⃗ M F ) T ] ⏟ [ I 4 ] Σ M F [ I 1 ] Σ M F [ I 2 ] Σ M F [ I 3 ] Σ M F [ I 4 ] Σ M F \begin{split} \left[ I \right] _{\Sigma _{\mathrm{M}}}^{F}\sum_i^N{m_{\mathrm{P}_{\mathrm{i}}}\cdot \left[ \left( \vec{R}_{\mathrm{P}_{\mathrm{i}}}^{F} \right) ^T\vec{R}_{\mathrm{P}_{\mathrm{i}}}^{F}\cdot E-\vec{R}_{\mathrm{P}_{\mathrm{i}}}^{F}\left( \vec{R}_{\mathrm{P}_{\mathrm{i}}}^{F} \right) ^T \right]} \\ \sum_i^N{m_{\mathrm{P}_{\mathrm{i}}}\cdot \left[ \left( \vec{R}_{\mathrm{M}}^{F}\left[ Q_{\mathrm{M}}^{F} \right] \vec{R}_{\mathrm{P}_{\mathrm{i}}}^{M} \right) ^{\mathrm{T}}\left( \vec{R}_{\mathrm{M}}^{F}\left[ Q_{\mathrm{M}}^{F} \right] \vec{R}_{\mathrm{P}_{\mathrm{i}}}^{M} \right) \cdot E-\left( \vec{R}_{\mathrm{M}}^{F}\left[ Q_{\mathrm{M}}^{F} \right] \vec{R}_{\mathrm{P}_{\mathrm{i}}}^{M} \right) \left( \vec{R}_{\mathrm{M}}^{F}\left[ Q_{\mathrm{M}}^{F} \right] \vec{R}_{\mathrm{P}_{\mathrm{i}}}^{M} \right) ^{\mathrm{T}} \right]} \\ \left\{ \begin{array}{c} \begin{array}{c} \underbrace{m_{\mathrm{total}}\cdot \left[ \left( \vec{R}_{\mathrm{M}}^{F} \right) ^{\mathrm{T}}\vec{R}_{\mathrm{M}}^{F}\cdot E-\vec{R}_{\mathrm{M}}^{F}\left( \vec{R}_{\mathrm{M}}^{F} \right) ^{\mathrm{T}} \right] }\\ \left[ I_1 \right] _{\Sigma _{\mathrm{M}}}^{F}\\ \end{array}\\ \begin{array}{c} \underbrace{\left[ Q_{\mathrm{M}}^{F} \right] \left( \sum_i^N{m_{\mathrm{P}_{\mathrm{i}}}\cdot \left[ \left( \vec{R}_{\mathrm{P}_{\mathrm{i}}}^{M} \right) ^{\mathrm{T}}\vec{R}_{\mathrm{P}_{\mathrm{i}}}^{M}\cdot E-\vec{R}_{\mathrm{P}_{\mathrm{i}}}^{M}\left( \vec{R}_{\mathrm{P}_{\mathrm{i}}}^{M} \right) ^{\mathrm{T}} \right]} \right) \left[ Q_{\mathrm{M}}^{F} \right] ^{\mathrm{T}}}\\ \left[ I_2 \right] _{\Sigma _{\mathrm{M}}}^{F}\\ \end{array}\\ \begin{array}{c} \underbrace{m_{\mathrm{total}}\cdot \left[ \left( \vec{R}_{\mathrm{M}}^{F} \right) ^{\mathrm{T}}\left( \left[ Q_{\mathrm{M}}^{F} \right] \vec{R}_{\mathrm{CoM}}^{M} \right) \cdot E-\vec{R}_{\mathrm{M}}^{F}\left( \left[ Q_{\mathrm{M}}^{F} \right] \vec{R}_{\mathrm{CoM}}^{M} \right) ^{\mathrm{T}} \right] }\\ \left[ I_3 \right] _{\Sigma _{\mathrm{M}}}^{F}\\ \end{array}\\ \begin{array}{c} \underbrace{m_{\mathrm{total}}\cdot \left[ \left( \left[ Q_{\mathrm{M}}^{F} \right] \vec{R}_{\mathrm{CoM}}^{M} \right) ^T\vec{R}_{\mathrm{M}}^{F}\cdot E-\left( \left[ Q_{\mathrm{M}}^{F} \right] \vec{R}_{\mathrm{CoM}}^{M} \right) \left( \vec{R}_{\mathrm{M}}^{F} \right) ^{\mathrm{T}} \right] }\\ \left[ I_4 \right] _{\Sigma _{\mathrm{M}}}^{F}\\ \end{array}\\ \end{array} \right. \\ \left[ I_1 \right] _{\Sigma _{\mathrm{M}}}^{F}\left[ I_2 \right] _{\Sigma _{\mathrm{M}}}^{F}\left[ I_3 \right] _{\Sigma _{\mathrm{M}}}^{F}\left[ I_4 \right] _{\Sigma _{\mathrm{M}}}^{F} \end{split} [I]ΣMFi∑NmPi⋅[(R PiF)TR PiF⋅E−R PiF(R PiF)T]i∑NmPi⋅[(R MF[QMF]R PiM)T(R MF[QMF]R PiM)⋅E−(R MF[QMF]R PiM)(R MF[QMF]R PiM)T]⎩ ⎨ ⎧ mtotal⋅[(R MF)TR MF⋅E−R MF(R MF)T][I1]ΣMF [QMF](i∑NmPi⋅[(R PiM)TR PiM⋅E−R PiM(R PiM)T])[QMF]T[I2]ΣMF mtotal⋅[(R MF)T([QMF]R CoMM)⋅E−R MF([QMF]R CoMM)T][I3]ΣMF mtotal⋅[([QMF]R CoMM)TR MF⋅E−([QMF]R CoMM)(R MF)T][I4]ΣMF[I1]ΣMF[I2]ΣMF[I3]ΣMF[I4]ΣMF 其中 [ I 2 ] Σ M F [ Q M F ] ( ∑ i N m P i ⋅ [ ( R ⃗ P i M ) T R ⃗ P i M ⋅ E − R ⃗ P i M ( R ⃗ P i M ) T ] ) [ Q M F ] T [ Q M F ] [ I ] Σ M M [ Q M F ] T \left[ I_2 \right] _{\Sigma _{\mathrm{M}}}^{F}\left[ Q_{\mathrm{M}}^{F} \right] \left( \sum_i^N{m_{\mathrm{P}_{\mathrm{i}}}\cdot \left[ \left( \vec{R}_{\mathrm{P}_{\mathrm{i}}}^{M} \right) ^{\mathrm{T}}\vec{R}_{\mathrm{P}_{\mathrm{i}}}^{M}\cdot E-\vec{R}_{\mathrm{P}_{\mathrm{i}}}^{M}\left( \vec{R}_{\mathrm{P}_{\mathrm{i}}}^{M} \right) ^{\mathrm{T}} \right]} \right) \left[ Q_{\mathrm{M}}^{F} \right] ^{\mathrm{T}}\left[ Q_{\mathrm{M}}^{F} \right] \left[ I \right] _{\Sigma _{\mathrm{M}}}^{M}\left[ Q_{\mathrm{M}}^{F} \right] ^{\mathrm{T}} [I2]ΣMF[QMF](∑iNmPi⋅[(R PiM)TR PiM⋅E−R PiM(R PiM)T])[QMF]T[QMF][I]ΣMM[QMF]T对上式进行讨论 纯回转 当 R ⃗ M F 0 \vec{R}_{\mathrm{M}}^{F}0 R MF0时化简为 [ I ] Σ M F ∣ R ⃗ M F 0 [ I 2 ] Σ M F [ Q M F ] ( ∑ i N m P i ⋅ [ ( R ⃗ P i M ) T R ⃗ P i M ⋅ E − R ⃗ P i M ( R ⃗ P i M ) T ] ) [ Q M F ] T [ Q M F ] [ I ] Σ M M [ Q M F ] T \left. \left[ I \right] _{\Sigma _{\mathrm{M}}}^{F} \right|_{\vec{\mathrm{R}}_{\mathrm{M}}^{F}0}\left[ I_2 \right] _{\Sigma _{\mathrm{M}}}^{F}\left[ Q_{\mathrm{M}}^{F} \right] \left( \sum_i^N{m_{\mathrm{P}_{\mathrm{i}}}\cdot \left[ \left( \vec{R}_{\mathrm{P}_{\mathrm{i}}}^{M} \right) ^{\mathrm{T}}\vec{R}_{\mathrm{P}_{\mathrm{i}}}^{M}\cdot E-\vec{R}_{\mathrm{P}_{\mathrm{i}}}^{M}\left( \vec{R}_{\mathrm{P}_{\mathrm{i}}}^{M} \right) ^{\mathrm{T}} \right]} \right) \left[ Q_{\mathrm{M}}^{F} \right] ^{\mathrm{T}}\left[ Q_{\mathrm{M}}^{F} \right] \left[ I \right] _{\Sigma _{\mathrm{M}}}^{M}\left[ Q_{\mathrm{M}}^{F} \right] ^{\mathrm{T}} [I]ΣMF R MF0[I2]ΣMF[QMF](i∑NmPi⋅[(R PiM)TR PiM⋅E−R PiM(R PiM)T])[QMF]T[QMF][I]ΣMM[QMF]T纯移动 当 R ⃗ M F ≠ 0 \vec{R}_{\mathrm{M}}^{F}\ne 0 R MF0且 [ Q M F ] E \left[ Q_{\mathrm{M}}^{F} \right] E [QMF]E时化简为 [ I ] Σ M F ∣ R ⃗ M F ≠ 0 , [ Q M F ] E [ I 1 ] Σ M F [ I ] Σ M M \left. \left[ I \right] _{\Sigma _{\mathrm{M}}}^{F} \right|_{\vec{\mathrm{R}}_{\mathrm{M}}^{F}\ne 0,\left[ Q_{\mathrm{M}}^{F} \right] \mathrm{E}}\left[ I_1 \right] _{\Sigma _{\mathrm{M}}}^{F}\left[ I \right] _{\Sigma _{\mathrm{M}}}^{M} [I]ΣMF R MF0,[QMF]E[I1]ΣMF[I]ΣMM 上式也称为惯性矩阵的平行轴定理Parallel Axis Theorem。运动坐标系原点与质心点重合 当 R ⃗ C o M F 0 \vec{R}_{\mathrm{CoM}}^{F}0 R CoMF0时化简为 [ I ] F ∣ R ⃗ C o M F 0 [ I 1 ] [ I 2 ] \left. \left[ I \right] ^F \right|_{\vec{R}_{\mathrm{CoM}}^{F}0}\left[ I_1 \right] \left[ I_2 \right] [I]F R CoMF0[I1][I2] 2.4 惯性矩阵的主轴定理} Principal Axis Theorem
进一步观察惯性矩阵 [ I ] M [ ∑ i N m P i ⋅ [ ( y P i M ) 2 ( z P i M ) 2 ] − ∑ i N m P i ⋅ x P i M y P i M − ∑ i N m P i ⋅ ( x P i M z P i M ) − ∑ i N m P i ⋅ ( y P i M x P i M ) ∑ i N m P i ⋅ [ ( x P i M ) 2 ( z P i M ) 2 ] − ∑ i N m P i ⋅ ( y P i M z P i M ) − ∑ i N m P i ⋅ ( z P i M x P i M ) − ∑ i N m P i ⋅ ( z P i M y P i M ) ∑ i N m P i ⋅ [ ( x P i M ) 2 ( y P i M ) 2 ] ] \left[ I \right] ^M\left[ \begin{matrix} \sum_i^N{m_{\mathrm{P}_{\mathrm{i}}}\cdot \left[ \left( y_{\mathrm{P}_{\mathrm{i}}}^{M} \right) ^2\left( z_{\mathrm{P}_{\mathrm{i}}}^{M} \right) ^2 \right]} -\sum_i^N{m_{\mathrm{P}_{\mathrm{i}}}\cdot x_{\mathrm{P}_{\mathrm{i}}}^{M}y_{\mathrm{P}_{\mathrm{i}}}^{M}} -\sum_i^N{m_{\mathrm{P}_{\mathrm{i}}}\cdot \left( x_{\mathrm{P}_{\mathrm{i}}}^{M}z_{\mathrm{P}_{\mathrm{i}}}^{M} \right)}\\ -\sum_i^N{m_{\mathrm{P}_{\mathrm{i}}}\cdot \left( y_{\mathrm{P}_{\mathrm{i}}}^{M}x_{\mathrm{P}_{\mathrm{i}}}^{M} \right)} \sum_i^N{m_{\mathrm{P}_{\mathrm{i}}}\cdot \left[ \left( x_{\mathrm{P}_{\mathrm{i}}}^{M} \right) ^2\left( z_{\mathrm{P}_{\mathrm{i}}}^{M} \right) ^2 \right]} -\sum_i^N{m_{\mathrm{P}_{\mathrm{i}}}\cdot \left( y_{\mathrm{P}_{\mathrm{i}}}^{M}z_{\mathrm{P}_{\mathrm{i}}}^{M} \right)}\\ -\sum_i^N{m_{\mathrm{P}_{\mathrm{i}}}\cdot \left( z_{\mathrm{P}_{\mathrm{i}}}^{M}x_{\mathrm{P}_{\mathrm{i}}}^{M} \right)} -\sum_i^N{m_{\mathrm{P}_{\mathrm{i}}}\cdot \left( z_{\mathrm{P}_{\mathrm{i}}}^{M}y_{\mathrm{P}_{\mathrm{i}}}^{M} \right)} \sum_i^N{m_{\mathrm{P}_{\mathrm{i}}}\cdot \left[ \left( x_{\mathrm{P}_{\mathrm{i}}}^{M} \right) ^2\left( y_{\mathrm{P}_{\mathrm{i}}}^{M} \right) ^2 \right]}\\ \end{matrix} \right] [I]M ∑iNmPi⋅[(yPiM)2(zPiM)2]−∑iNmPi⋅(yPiMxPiM)−∑iNmPi⋅(zPiMxPiM)−∑iNmPi⋅xPiMyPiM∑iNmPi⋅[(xPiM)2(zPiM)2]−∑iNmPi⋅(zPiMyPiM)−∑iNmPi⋅(xPiMzPiM)−∑iNmPi⋅(yPiMzPiM)∑iNmPi⋅[(xPiM)2(yPiM)2] 为对称矩阵Symmetric Matrix此时默认 M M M 点与 F F F 点重合则一定能够对角化。
等价于找到另一原点与 M M M 重合的坐标系 B B B 使得 [ I ] B [ I x x B 0 0 0 I y y B 0 0 0 I z z B ] \left[ I \right] ^B\left[ \begin{matrix} I_{\mathrm{xx}}^{B} 0 0\\ 0 I_{\mathrm{yy}}^{B} 0\\ 0 0 I_{\mathrm{zz}}^{B}\\ \end{matrix} \right] [I]B IxxB000IyyB000IzzB 根据矩阵对角化Matrix Diagonalizing的原理结合纯回转推导可得 [ I ] M [ Q B M ] [ I ] B [ Q B M ] T \left[ I \right] ^M\left[ Q_{\mathrm{B}}^{M} \right] \left[ I \right] ^B\left[ Q_{\mathrm{B}}^{M} \right] ^{\mathrm{T}} [I]M[QBM][I]B[QBM]T 其中 [ Q B M ] \left[ Q_{\mathrm{B}}^{M} \right] [QBM] 满足 [ i ⃗ B j ⃗ B k ⃗ B ] [ Q B M ] T [ i ⃗ M j ⃗ M k ⃗ M ] \left[ \begin{array}{c} \vec{i}^B\\ \vec{j}^B\\ \vec{k}^B\\ \end{array} \right] \left[ Q_{\mathrm{B}}^{M} \right] ^{\mathrm{T}}\left[ \begin{array}{c} \vec{i}^M\\ \vec{j}^M\\ \vec{k}^M\\ \end{array} \right] i Bj Bk B [QBM]T i Mj Mk M ( I x x B , I y y B , I z z B ) \left( I_{\mathrm{xx}}^{B},I_{\mathrm{yy}}^{B},I_{\mathrm{zz}}^{B} \right) (IxxB,IyyB,IzzB) 为矩阵 [ I ] M \left[ I \right] ^M [I]M的特征值Eigenvalue [ Q B M ] \left[ Q_{\mathrm{B}}^{M} \right] [QBM] 为对应于特征值矩阵 [ I ] B \left[ I \right] ^B [I]B的特征基Standard Eigenvalue Basis(列向量)
